Updated Lagrangian form

By integrating each term of Nanson s relation nds = J F N dS we have 

Equation 2.128 implies that II is an image of n in the deformed body mapped from the undeformed body by J F. A component form of (2.128) is given by 

Thus the partial differential equation system of the updated Lagrangian form together with the boundary conditions is given by 

It must be noted that, as understood from (2.131), the updated Lagrangian form is expressed in Eulerian terms . 

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For nonlinear problems such as elasto-plastic materials it is necessary to use a formulation based on an incremental form of the equation of equilibrium. We can introduce either the total Lagrangian form or the updated Lagrangianform. In the former case the incremental form is expressed in Lagrangian terms, while in the latter case the incremental form is given in an Eulerian description. 

Incremental Form of the Updated Lagrangian Equation of Motion... 

For the deformed body (i.e., Eulerian description) we can introduce a Legendre transformation it should be noted that material isotropy is imposed. In the analysis of finite strain problems the updated Lagrangian equilibrium equation, which is a form of an Eulerian description (cf. Sect. 2.6.2) is usually used, meaning that the body is required to be isotropic. 

Every detached bubble enters the electrolyte and a Lagrangian tracking procedure is used to update the velocities and positions of all dispersed gas bubbles in the electrolyte at each time step of the Navier-Stokes solver. From Newton s second law, an equation of motion can be obtained for every bubble, based on the formulation stated in [24], Together with the relation between the particle s position and velocity, a set of two ordinary differential equations in three space dimensions can be formed in order to update the bubble trajectory... 

In stochastic Lagrangian particle models, the evolution of the concentration field is computed in a two-step process. First, the Eulerian velocity field in the region of interest must be calculated, either by solution of the Navier-Stokes equations or via an approximate method that satisfies mass consistency. The solution must also provide the local statistics of the velocity field. Individual particles are then released, and their position is updated over a time increment dt using an equation of the form (Wilson and Sawford 1996)... 

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